The grounding electrode conductor connects the service’s grounded conductor to the earth electrode system, and it answers different physics than its cousin the EGC: the GEC handles lightning and surge gradients, not breaker-tripping fault current. It is sized from the largest ungrounded service-entrance conductor per NEC Table 250.66 — a typical 200 A residential service on 2/0 or 3/0 copper lands in the table’s third row, a 4 AWG copper or 2 AWG aluminum GEC. Parallel service sets are handled by the table note on equivalent area: add the circular-mil areas on one phase, so two 4/0 copper conductors per phase are 423,200 cmil, the "over 350 through 600 kcmil" row, and the GEC is 1/0 copper even though each individual conductor is only 4/0.
Three reliefs in 250.66(A)–(C) cap the size by electrode type: a GEC that is the sole connection to a rod, pipe, or plate electrode never needs to be larger than 6 AWG copper — the reason 6 AWG bare copper to the ground rods is so common — a concrete-encased electrode caps at 4 AWG copper, and a ground ring follows its ring conductor. Table lookups are a code starting point rather than the whole check: installation conditions and local amendments can change the answer, and the full Article 250 review still applies.